A377130 -id:A377130 - cp's OEIS frontend

A038119 Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).

1, 1, 2, 7, 23, 112, 607, 3811, 25413, 178083, 1279537, 9371094, 69513546, 520878101, 3934285874, 29915913663, 228779330204, 1758309223457, 13573319825615, 105192814197984, 818136047201932, 6383528588447574

Offset: 1

N. J. A. Sloane

a(1)-a(12) computed by Achim Flammenkamp.
A000162 but with one copy of each mirror-image deleted.
From R. J. Mathar, Mar 19 2018: (Start)
We can split the numbers into an irregular table which lists in row n how many configurations have c contacts for c >= 0:
  1;
  0 1;
  0 0 2;
  0 0 0 6 1;
  0 0 0 0 21 2;
  0 0 0 0 0 91 19 2;
  0 0 0 0 0 0 484 110 12 1;
  0 0 0 0 0 0 0 2817 852 129 12 0 1;
  0 0 0 0 0 0 0 0 17788 6321 1166 132 5 1;
Row lengths are 1+A007818(n). Row sums are a(n).
(End)
Number of unoriented polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}. For unoriented polyominoes, chiral pairs are counted as one.- Robert A. Russell, Mar 21 2024

S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition (Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.
W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972. [See https://books.google.nl/books?id=ja7iBQAAQBAJ&pg=PA101]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Achim Flammenkamp, Home page.
Kevin L. Gong, Polyominoes Home Page.
John Mason, Counting free polycubes.
John Mason, Coordinate sets of examples of polycube symmetry (version 2)

A038119 = (A007743+A000162)/2, A007743 = 2*A038119 - A000162, A000162 = 2*A038119 - A007743.
32nd row of A366766.
Cf. A000162 (oriented), A371397 (chiral), A007743 (achiral), A001931 (fixed).
Cf. for each symmetry: A376964, A376965, A376966, A376967, A376968, A376969, A376970, A376972, A376973, A376974, A376975, A376976, A376977, A376978, A376979, A376980, A376981, A376982, A376983, A377127, A376984, A376985, A376986, A376987, A376988, A376989, A377128, A376990, A376991, A377129, A377130, A377131, A376971

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A000162 = A@000162;
A007743 = A@007743;
a[n_] := (A007743[[n]] + A000162[[n]])/2;
a /@ Range[16] (* Jean-François Alcover, Jan 16 2020 *)

a(n) = A000162(n) - A371397(n) = A371397(n) + A007743(n). - Robert A. Russell, Mar 21 2024

More terms from Brendan Owen (brendan_owen(AT)yahoo.com), Jan 02 2002
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
More terms from John Mason, Sep 19 2024

A007743 Number of achiral polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4} (or polycubes).

Original entry on oeis.org

1, 1, 2, 6, 17, 58, 191, 700, 2515, 9623, 36552, 143761, 564443, 2259905, 9057278, 36705846, 149046429, 609246350, 2495727647, 10267016450, 42322763940, 174974139365

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

A000162 but with both copies of each mirror-image deleted.

An achiral polyomino is identical to its reflection. Many of these achiral polyominoes do not have a plane of symmetry. For example, the hexomino with cell centers (0,0,0), (0,0,1), (0,1,1), (1,1,1), (1,2,1), and (1,2,2) has a center of symmetry at (1/2,1,1) but no plane of symmetry. The decomino with cell centers (0,0,0), (0,0,1), (0,1,1), (0,2,1), (0,2,2), (1,0,2), (1,1,2), (1,1,1), (1,1,0), and (1,2,0) has no plane or center of symmetry. - Robert A. Russell, Mar 21 2024

G. Thimm, Emails to N. J. A. Sloane, Sep. 1994

A038119 = (A007743+A000162)/2, A007743 = 2*A038119 - A000162, A000162 = 2*A038119 - A007743.
Cf. A000162 (oriented), A038119 (unoriented), A371397 (chiral), A001931 (fixed).
Component symmetries: A376969, A376970, A376971, A376972, A376973, A376974, A376977, A376978, A376979, A376980, A376981, A376983, A376984, A376985, A376986, A376987, A376988, A376989, A376990, A376991, A377129, A377130.

a(n) = A000162(n) - 2*A371397(n) = A038119(n) - A371397(n). - Robert A. Russell, Mar 21 2024

a(13)-a(16) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
Changed "symmetric" to "mirror-symmetric" in the title by George Sicherman, Feb 21 2018
Changed "mirror-symmetric" to "achiral" in the title to ensure that a plane of symmetry is not required. - Robert A. Russell, Mar 21 2024
a(17)-a(22) from John Mason, Sep 19 2024

A377155 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 12.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 1, 1, 3, 4, 0, 8, 4, 3, 8, 3, 2, 17, 6, 8, 19, 27, 2, 53, 19, 26, 49, 19, 10, 127, 38, 64, 121, 166, 15, 373, 111, 197, 306, 150, 67, 923, 242, 460, 771, 1100, 115, 2665, 686, 1405, 1972, 1085, 431, 6681, 1562, 3335, 5051, 7353

Offset: 1

John Mason, Oct 18 2024

nonn

The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice.

W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

Cf. A000162, A066273, A066281, A066283, A066287, A066288, A066453, A066454, A377128, A377129, A377130.

a(n) = 2*BD(n)+CCC(n)+DEE(n) = 2*A377128(n)+A377129(n)+A377130(n); see Lunnon paper for naming convention.

cp's OEIS Frontend

A038119 Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).

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A007743 Number of achiral polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4} (or polycubes).

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A377155 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 12.

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